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Description
Cortical bone is an example of a multiscale biological tissue, being characterized by a sharp separation between the macroscopic lenght scale, at which the cortical bone appears like a continuum with defined structural biological functionality, and the microscopic length scale, at which it is possible to identify an elementary unit, i.e. the lamella, that, when repeated, composes the tissue. Hence, the macroscopic mechanical behaviour of the cortical bone depends on the resolution of microscopic processes such as remodeling, but also the converse is true. In particular, we study the reorganization of the bone’s inner structure that leads to the formation of “plastic zones”, which are diffuse plastic interfaces covering multiple lamellae [1].
In our work, we put forward a general framework for studying the mechanics of a composite medium, made of two solids constituents, both of which can incur remodelling, and that present a microscale characterized by a unitary cell of arbitrary shape. In order to study the formation of a diffuse interface, we adapt Gurtin&Anand theory of strain-gradient plasticity [2] to this biological setting. Under the hypothesis of well-separation of the geometrical scales, we employ the techniques of Asymptotic Homogenization to derive the homogenized response of the system in the form of effective coefficients embedding the microscopic structural information. Lastly, we perform some numerical simulations on a simplified geometrical setting to provide an evaluation of the strain gradient remodeling effects [3].
References:
[1] D. M. Robertson, D. Robertson, C. R. Barrett, Journal of Biomechanics, Elsevier BV, 1978.
[2] M. E. Gurtin, L. Anand, International Journal of Plasticity, Elsevier BV, 2005.
[3] A. Giammarini, A. Ramírez-Torres, A. Grillo, Math. Methods in the App. Sciences, Wiley, 2024.